Optimal. Leaf size=11 \[ \frac{x}{\sqrt{x^2+1}} \]
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Rubi [A] time = 0.0011778, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {191} \[ \frac{x}{\sqrt{x^2+1}} \]
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (1+x^2\right )^{3/2}} \, dx &=\frac{x}{\sqrt{1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0020271, size = 11, normalized size = 1. \[ \frac{x}{\sqrt{x^2+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 10, normalized size = 0.9 \begin{align*}{x{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948197, size = 12, normalized size = 1.09 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.420945, size = 53, normalized size = 4.82 \begin{align*} \frac{x^{2} + \sqrt{x^{2} + 1} x + 1}{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.694002, size = 8, normalized size = 0.73 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07279, size = 12, normalized size = 1.09 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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